On diophantine monoids and their class groups

Abstract

A Diophantine monoid S is a monoid which consists of the set of solutions in nonnegative integers to a system of linear Diophantine equations. Given a Diophantine monoid S, we explore its algebraic properties in terms of its defining integer matrix A. If d r (S) and d c (S) denote respectively the minimal number of rows and minimal number of columns of a defining matrix A for S, then we prove in Section 3 that d r (S) = rank Cl(S) and d c (S) = rank Cl(S)+rank Q(S) where Cl(S) represents the divisor class group of S and Q(S) the quotient group of S. The proof relies on the characteristic properties of the so-called essential states of S, which are developed in Section 2. We close in Section 4 by offering a characterization of factorial Diophantine monoids and an algorithm which determines if a Diophantine monoid is half-factorial.